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Subsections

5 Rotational velocities

5.1 Methodology

In order to derive rotational velocities we attempted to fit Gaussian profiles to 4 HeI lines at 4026 Å, 4143 Å, 4387 Å and 4471 Å in the spectra. This was successful for most of the objects, although for a small number only 2 or 3 lines could be reliably fitted due to contamination by nearby emission or absorption features. The profile full widths at half maximum were converted to $v \sin i$ using a fit to the 4471 Å full width half maximum - $v \sin i$ correlation of Slettebak et al. (1975). Making the appropriate correction for the differing central wavelengths of each line, the fits employed were:
\begin{displaymath}
v \sin i = 41.25 F(4471) {\rm\ km~s^{-1}}\end{displaymath} (1)

\begin{displaymath}
v \sin i = 42.03 F(4387) {\rm\ km~s^{-1}}\end{displaymath} (2)

\begin{displaymath}
v \sin i = 44.51 F(4143) {\rm\ km~s^{-1}}\end{displaymath} (3)

\begin{displaymath}
v \sin i = 45.82 F(4026) {\rm\ km~s^{-1}}\end{displaymath} (4)
where $F(\lambda)$ is the full width half maximum in Å at a wavelength of $\lambda$ Å. The $v \sin i$ quoted in Table 4 is the mean of those derived from all of the fitted lines for each object after correction for the mean instrumental velocity dispersion of 55 km s-1 (determined from measurements of interstellar lines in the spectra). The errors reflect the dispersion in the measured full width half maxima, with the minimum error set at 10 km s-1.

26 of the objects in our sample have previously measured $v \sin i$'s in the compilations of Bernacca & Perinotto (1970, 1971). A comparison of the historical $v \sin i$'s with those we derive shows the historical values typically $\sim 20$ per-cent greater than our values. However Bernacca & Perinotto (1974) state that their $v \sin i$'s are referenced to the scale of Slettebak (1968) whereas our measurements are instead referenced to Slettebak et al. (1975). Figure 7 of that paper shows that the new (1975) scale derives $v \sin i$'s some 15-20 per-cent smaller for B stars than the old (1968) scale and so the discrepancy may simply be understood to be caused by our use of a more modern $v \sin i$ calibration. To make this clear Fig. 7 re-plots Fig. 7 of Slettebak et al. (1975) with their standard stars marked as squares and our sample marked as crosses. No significant difference is apparent between the two distributions.

  
\begin{figure}
\setlength {\unitlength}{1.0in}
 
\begin{picture}
(3.0,3.0)(0,0)
\put(-0.6,-0.5){\epsfbox[0 0 2 2]{h1180f7.ps}}\end{picture}\end{figure} Figure 7: Comparison of "old'' (Slettebak 1968) and ``new'' (Slettebak 1975) $v \sin i$ scales for B stars (square symbols) and our measurements of Be stars (new scale) versus Bernacca & Perinottot's (1970, 1971) (old scale) data for the same objects (star symbols)

5.2 Distribution of rotational velocities within the sample

In Fig. 8 (hollow plus filled areas) we plot the distribution of $v \sin i$ versus spectral type within the sample for luminosity classes III, IV and V. It is important to identify whether there are are biases in the $v \sin i$ values within the sample. There are two effects that may lead to a relationship between $v \sin i$ and brightness for Be stars and in a flux limited sample we would of course expect an inherent bias towards intrinsically brighter objects.

The first effect that would lead to a bias towards rapidly rotating Be stars is described by Zorec & Briot (1997). The more rapidly rotating stars will suffer greater deformation than the slower rotators. Although the bolometric luminosity from an object is of course conserved, the effect of rotation is to make the spectrum appear cooler. Since the peak of B star spectra is in the UV, this will make the optical flux brighter for most aspect ratios (Collins et al. 1991; Porter 1995), leading to the preferential selection of more rapidly rotating stars in a magnitude limited sample.

In addition it is well known that the emission produced in the circumstellar envelope of Be stars leads to an increase in their optical flux. Observations of phase changes from non-Be to Be typically show increases of 0.1-0.2 magnitudes (Feinstein 1975; Apparao 1991). This is due to reprocessing of the UV radiation from the underlying Be star into optical and infrared light by the disk. Assuming a relation between disk size and excess optical flux we would therefore expect stars with larger circumstellar disks to be preferentially selected by our flux limited sample. Also assuming that the sizes of Be star disks are likely to be sensitive to the stellar rotational velocity, then we would expect the most rapidly rotating stars to show the strongest optical excess. This again would lead to a bias towards rapidly rotating Be stars in a magnitude limited sample.

  
\begin{figure}
\setlength {\unitlength}{1.0in}
 
\begin{picture}
(3.0,6.1)(0,0)
\put(-0.0,-0.0){\epsfbox[0 0 2 2]{h1180f8.ps}}\end{picture}\end{figure} Figure 8: $v \sin i$ distributions for all objects in the sample (hollow plus filled areas) and objects in a volume limited subset of the sample (filled area only)

  
\begin{figure}
\includegraphics [width=7.3cm]{h1180f9.ps}\end{figure} Figure 9: Apparent B magnitude versus B spectral subclass for the volume limited sample. Symbols as per Fig. 6

In order to test whether our sample is biased in this way we must compare it to a volume limited subset. This can be created from our sample by using the absolute magnitudes (Schmidt-Kaler 1982) derived from the spectral and luminosity classes to select objects that lie within the volume defined by the absolute magnitude limit for the intrinsically faintest objects (B9V) at the apparent magnitude limit ($B \sim 11$). The resulting volume limited subsample contains 34 objects out of our original 58. We plot the B magnitude distribution of this subsample in Fig. 9. Note how the volume limiting naturally cuts out the objects with faint apparent magnitudes at early spectral types (cf. Fig. 6). The $v \sin i$ distribution of the volume limited subsample is plotted in Fig. 8 (filled area only) to allow comparison with the total sample (filled plus hollow area). In order to compare the distributions we use a Kolmogorov-Smirnov (KS) test between the volume-limited and total samples. This shows that the probability of the distribution of $v \sin i$ two samples being the same is 80% for luminosity class III, 99% for luminosity class IV, and 95% for luminosity class V. There is therefore no statistical evidence for any bias in the $v \sin i$ values for all three luminosity classes in the sample. What is clear from Fig. 8 is that there is a considerably lower mean $v \sin i$ for luminosity class III as opposed to class V Be stars. The astrophysical interpretation of this result is discussed in Steele (1999).


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